Projects

Wilmott Magazine about Broda

The British-Russian Offshore Development Agency (Broda) has implemented Sobol'
sequence quasi-random generation software up to and including dimension 370.
Russian mathematical guru, Ilya Sobol' pioneered this sequence and modified
direction numbers for it. Broda chief co-ordinator described Sobol' as: "A
theorist who has always worked very closely with practitioners from the 1960s
onwards and has been involved with many big projects - he has made incredible
contributions to the field of applied quantitive mathematics." The quasi-random
numbers are used for pricing derivatives and for value at risk in the form of
highly efficient high-dimensional Monte Carlo simulations. This latest development
from Broda has been described by the organization as evidence of their support
for all areas of information systems. In an official statement it said: "This
is further evidence of us bringing Russian mathematical expertise and scientific
skills to the West."

Enhanced 32000-dimensional Sobol' LDS generator SobolSeq32000

BRODA has been for many years a leader in the development and distribution
of the high dimensional LDS generators. One of its well known and popular products
is 370 dimensional Sobol' LDS generator SobolSeq370 developed by Prof. I. Sobol'.
Over the last few years SobolSeq370 has become an industry standard in many fields
of mathematics and finance. This is what Peter Jackel who is one of the leading
experts in the field of Monte Carlo methods in finance says in his book about
SobolSeq370: "There is a commercial library module available from an organisation
called BRODA that can generate Sobol' sequences in up to 370 dimensions. In a way,
this module can claim to be a genuine Sobol' number generator since Professor Sobol'
himself is behind the initialisation numbers that drive the sequence, and he is
also linked to the company distributing the library." (Peter Jackel, "Monte Carlo
methods in finance", John Wiley & Sons, 2002.) With constantly increasing complexity
of problems there has been a need for very dimensional LDS generators. BRODA has
recently developed a new 32000 dimensional Sobol' LDS generator SobolSeq32000
for maximum dimension 32000 which
has even better performance and efficiency than previously developed
generators. Not only this generator has
very high dimensionality and employs the super fast generation
algorithm but the generated Sobol' sequences satisfy Property A in
all dimensions and property A' for the adjacent dimensions.
BRODA's SobolSeq generators outperform all other known
generators both in speed and accuracy as shown in this paper:
I. Sobol’, D. Asotsky, A. Kreinin, S. Kucherenko. Construction and
Comparison of High-Dimensional Sobol’ Generators, 2011, Wilmott
Journal, Nov, pp. 64-79, 2012), which can be downloaded
here.

Global Sensitivity Analysis of nonlinear models

Global Sensitivity Analysis (GSA) quantifies the relative importance of input model
parameters (variables) in determining the value of the output variable. GSA enables
to identify key parameters whose uncertainly affects most of the output. It then
can be used to rank variables, fix unessential variables or decrease dimensionality
of the problem. I. Sobol' developed the most general GSA method introducing global
sensitivity indices. This method is an ideal tool for the analysis of complex
multidimensional nonlinear models. Review on global GSA
(Sobol' I., Kucherenko S. "Global Sensitivity Indices for Nonlinear Mathematical
Models. Review", Wilmott Magazine, 2005, Vol. 2) can be downloaded
here.
The most recent paper on application of GSA in Monte Carlo Option Pricing
(Kucherenko S., Shah N. "The Importance of being Global. Application of Global Sensitivity Analysis in
Monte Carlo option Pricing", Wilmott Magazine, 2007, Vol. 4) can be found
here. See also a slide presentation on
Application of GSA in Monte Carlo and Quasi Monte Carlo option Pricing
here.
Four different types of options and Greeks are considered. It is shown that the efficiency of Quasi Monte Carlo
changes with the change of effective dimensions and it can be different not only for different payoffs
but even for different Greeks for the same payoff.
Methods and techniques of GSA and the difference between the nominal and
effective dimensions, which is very important to understand the superior
performance of Sobol' sequences even in very high dimensional problems are presented
in this
paper.

Many problems of financial mathematics and risk analysis can be formulated as
global nonlinear optimization problems (GNOP). Despite of huge practical
importance of GNOP, problem of developing robust and efficient numerical
methods for GNOP is far from being solved. Existing methods have either limited
applicability or are inefficient because of prohibitively large required CPU-time
and therefore have limited practical usefulness. There are two kinds of commonly
used techniques for solving GNOP: deterministic and stochastic. Deterministic
methods guarantee convergence to a global solution. However, for large-scale
problems these methods require prohibitively large CPU-time. And they are
applicable only if the objective function and the constraints are twice
continuously differentiable. Stochastic methods are much faster than deterministic
methods. They can be used for any class of objective functions Stochastic search
methods yield an asymptotic (in a limit N, where N are randomly chosen points)
guarantee of convergence. But as in reality problems can be solved with limited
sets of sample points N, convergence to a global solution is not guaranteed.

Broda developed a novel deterministic method, which combines advantages of
deterministic and Stochastic methods. It is based on application of low-discrepancy
sequences (LDS) and multi-level linkage methods. In comparison with Stochastic
methods employing random numbers application of LDS significantly decreases
number of points N required to achieve the same tolerance of finding the global
minimum. At the same time it is a deterministic method. It has deterministic error
bounds instead of probabilistic Monte Carlo error bounds. Many studies have shown
that Sobol' LDS is superior to other known LDS and it is used in the developed
method. To learn more, please download these
papers.